Metamaterials based on resonant-cell structures are often used to obtain a negative index of refraction, where both the effective permittivity and permeability are negative, and operation is adjusted to occur just above spectrally overlapping magnetic and electric resonances of the cell structures. See S. Tretyakov, Analytical Modeling in Applied Electromaqnetics, Artech House (2003). Other applications of metamaterials, including cloaking, require independent tuning of the permittivity and permeability and thus require the ability to place the resonances at the desired spectral positions. In addition to the selective placement of resonances, remaining in the effective material limit (with diffraction suppressed) is another goal in these types of metamaterial applications. See D. R. Smith and J. B. Pendry, J. Opt. Soc. Am. B 23(3), (2006); J. M. Lerat et al., J. Appl. Phys. 100, 084908 (2006); R. Liu et al., Phys. Rev. E 76, 026606 (2007); Th. Koschny et al., Phys. Rev. B 71, 245105 (2005); and P. A. Belov and C. R. Simovski, Phys. Rev. E 72, 026615 (2005). Often times unit cells containing metallic split-ring resonators (yielding the magnetically-resonant component) and loaded dipoles (yielding the electrically-resonant component) are used in constructing negative-index metamaterials since they can be small and still attain both negative permittivity and permeability. See J. B. Pendry et al., IEEE Trans. Microwave Theory Tech. 47(11), (1999); S. Tretyakov et al., IEEE Trans. Antennas Propag. 51, 2562 (2003); S. Tretyakov, Microwave and Optical Technology Letters 31(3), 163 (2001); J. Kim and A. Gopinath, Phys. Rev. B 76, 115126 (2007); B. Popa and S. Cummer, Phys. Rev. Lett. 100, 207401 (2008); M. Sinclair et al., SPIE Optics+Photonics (2011); and L. I. Basilio et al., IEEE Antennas Wireless Propag. Lett. 10, 1567 (2011). In these cases, tuning of the electric and magnetic resonances is achieved through the design of the respective resonators.
In recent years, all-dielectric resonant structures utilizing high-permittivity materials have drawn much attention since their use eliminates the material absorption inherent to metallic structures (which can become prohibitive at higher frequencies). While dielectric cylinders and rectangles have frequently been used to realize a medium of negative-permeability, all-dielectric resonators are not a natural fit for negative-index applications, since the first magnetic resonance occurs at a lower frequency than the first electric resonance. See J. Kim and A. Gopinath, Phys. Rev. B 76, 115126 (2007); B. Popa and S. Cummer, Phys. Rev. Lett. 100, 207401 (2008); and M. Sinclair et al., SPIE Optics+Photonics (2011).
Well-known approaches that have been used to attempt to align the resonances of all-dielectric resonators include the core-shell designs of Kuester and Basilio and the AB-type designs of Ahmadi and Jylha. See E. Kuester et al., Prog. Electromag. Res. B 33, 175 (2011); L. Basilio et al., IEEE APS-Symp., Spokane, Wa., USA (2011); A. Ahmadi and H. Mosallaei, Phys. Rev. B 77, 045104 (2008); and L. Jylha et al., J. Appl. Phys. 99, 043102, 2006. While both of these methods introduce an additional degree of freedom that provides for the tuning of the resonances (in Kuester and Basilio by introducing a surrounding dielectric shell layer to a dielectric core and in Ahmadi and Jylha by introducing an additional resonator particle into the unit cell), unfortunately these two approaches can easily bring into question the applicability of effective media; this becomes particularly apparent at higher operating frequencies. In the case of the AB-type design, the size of the unit cell is physically extended (perhaps by a factor of two) to accommodate the additional resonator while, alternatively, in the core-shell design the electrical size of the resonator is forced to increase because overlap of only higher-order modes is possible. As the operating frequency is increased, an additional problem that arises in both these approaches (and any other all-dielectric designs) is that the range of available permittivities becomes much more limited. For example, in the long-wave infrared (8 μm-15 μm) the largest relative permittivities available in low-loss dielectric materials are in the range of 25-32, while relative permittivities in the hundreds (or higher) are common in the radio frequency part of the spectrum. See E. Palik, Handbook of Optical Constants and Solids, Academic, Orlando, Fla., (1986). To achieve resonance and still remain in the effective medium limit then becomes a difficult proposition. Nevertheless, as metamaterials designs are pushed to higher frequencies, the need for dielectric resonators is imperative since absorption associated with the metallic resonators becomes significant.
Therefore, a need remains for metamaterials based on dielectric resonators where degeneracy of the lowest-order magnetic- and electric-resonant modes can be realized and which consequently do not increase the lattice spacing.